Re: Inconsistent behaviour of Integrate

*To*: mathgroup at smc.vnet.net*Subject*: [mg112424] Re: Inconsistent behaviour of Integrate*From*: Andrzej Kozlowski <akoz at mimuw.edu.pl>*Date*: Wed, 15 Sep 2010 04:38:14 -0400 (EDT)

This depends on the speed on your computer. On my MacBook PRI I get: In[1]:== Integrate[ Sqrt[(x - 1/2)^2 + (y - 1/2)^2], {x, 0, 1}, {y, 0, 1}] Out[1]== (1/24)*(4*Sqrt[2] + Log[17 + 12*Sqrt[2]]) However, if I use Maxim Rytin's trick to reduce performance thus: Dynamic[Pause[.5], UpdateInterval -> 1] ClearSystemCache[] Then I get: In[5]:== Integrate[ Sqrt[(x - 1/2)^2 + (y - 1/2)^2], {x, 0, 1}, {y, 0, 1}] Out[5]== (1/6)*(Sqrt[2] + ArcSinh[1]) Andrzej Kozlowski On 14 Sep 2010, at 11:12, Andreas Maier wrote: > Hello, > > I'm using Mathematica 7.0.1.0 on Linux x86 (64bit). I have a notebook > file, where I integrate the same integral twice: > > In[1]:== Integrate[Sqrt[(x - 1/2)^2 + (y - 1/2)^2], {x, 0, 1}, {y, 0, > 1}] > Out[1]== 1/6 (Sqrt[2] + ArcSinh[1]) > > In[2]:== Integrate[Sqrt[(x - 1/2)^2 + (y - 1/2)^2], {x, 0, 1}, {y, 0, > 1}] > Out[2]== 1/24 (4 Sqrt[2] + Log[17 + 12 Sqrt[2]]) > > As you can see from the output, integrating the same integral a second > time gives a different result. If I integrate the same integral a > third and a fourth time I always get the second result again. Only if > I restart the mathematica kernel, I get the first result again. > The results are equivalent, since > > Log[17 + 12 Sqrt[2]] == Log[(1 + Sqrt[2])^4] == 4* Log[(1 + Sqrt[2]) == 4= * ArcSinh[1] > > but somehow Mathematica seems to be able to do this simplification > only once. Is this inconsistent behaviour a bug? Is there a > possibility to give mathematica a hint, so that he always find the > first solution 1/6 (Sqrt[2] + ArcSinh[1]) to the integral? > From > > In[3]:== Expand[(1 + Sqrt[2])^4] > Out[3]== 17 + 12 Sqrt[2] > > In[4]:== Factor[%] > Out[4]== 17 + 12 Sqrt[2] > > I also figured that Mathematica doesn't seem to be able to factorize > an expression like 17 + 12 Sqrt[2] into (1 + Sqrt[2])^4. Is this a > known problem? Or should I use a different command to find this > factorization? > > Sincerely, > Andreas Maier >